Exploring Geography Cartographically

01/12/2001


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Table of Contents

Exploring Geography Cartographically

Subjects To Be Covered

To begin I stress

There are many ways of specifying location

Other Ways of Determining Locations

Common Geographic Locational Aliases and Conversions

Hand Held GPS with Map Locations are estimated from distances

Wrist Watch GPS

The fact that locations can be determined by specifying distances can lead to some interesting results.

In the lower school grades one learns the procedure for finding distance from coordinates.

An Exercise Using Distances

A Similar, But Simpler, Exercise

Solution Procedure

Step by Step Trilateration

Trilateration Procedure Illustrated Graphically

The Beginning Steps

The Answer, with Coordinates and Estimated Distances

Solution to the US Distance Exercise with (enlarged) estimated standard errors

Locations increase linearly, distances quadraticaly thus No Numbers are Needed!

I will now give you another example where locations are estimated from distances

Cappadocian Cuneiform Tablet

Cuneiform Tablet

Given the occurrence of town names on tablets

Use the well known gravity model:

Predicted Locations of 33 Pre-Hittite Towns in Cappadocia using distances estimated from presumed interaction.

USA Highway Distance Map

PPT Slide

This “distorted” map can be evaluated by the methods of Tissot

All maps are of course distorted to some extent

The Surface of the Earth Is Two-dimensional

The Map Projection Problem is to reduce the curved, closed, and bumpy two dimensional surface of the earth to a flat two dimensional surface

The Transform - Solve - Invert Paradigm

The conventional satellite tracking chart

Breckman Chart to Track Satellites

PPT Slide

PPT Slide

PPT Slide

US Population By One Degree Quadrilaterals

The Lat/Lon Grid in the Two Spaces Left, the usual grid. Right, transformed according to population.

US Map in the Two Spaces Left, the usual map. Right, the transform.

The Inversion On the right are uniform hexagons in the transformed space. On the left is the solution: The inverse transformation partitions the US into cells of equal population

Most of you are familiar with linear regression and correlation

Title page from Ptolemy’s Geography

A Page from Ptolemy’s Geography

Ptolemy’s World Map

Ptolemy’s Map of Gaul (Modern France)

Modern and Ptolomaic Latitude and Longitude for Gaul

The data in the foregoing table were assembled by a student and some of you may be more familiar with the the ancient names than I.

PPT Slide

Modern Latitude versus Ptolemy’s Latitude

Modern Longitude versus Ptolemy’s Longitude

PPT Slide

Ptolemaic Displacements Amounts needed to move Ptolemy’s locations to modern locations. The places needing further investigation can be seen.

Interpolated Vector Field Amount needed to move the graticule to modern locations

Ptolemaic Grid Shifted to Fit Modern ‘Pushed’ by the interpolated vector field

There are many applications of vectors (pairs of coordinates) in geography

Geographical Interpolation

But often we have observations assembled by statistical units Census tracts, school districts, and the like

Pycnophylactic Reallocation

Population Density by County

Population Density in Kansas By County Courtesy of T. Slocum

Population Density in Kansas by County Each county still contains the same number of people

How Pycnophylactic Reallocation Works

Mass Preserving Reallocation Using Areal Data

What the Mathematics Means

Colored Clay Before Smoothing

Colored Clay After Smoothing

Another Advantage of Mass-Preserving Reallocation

The next important topic is

The Table is an Important Form of Geographic Movement Data

Let T represent the table, with i rows and j columns. It can be decomposed into two parts as follows: Tij = T+ + T– where T+ = (Tij + Tji)/2 (symmetric) T– = (Tij - Tji)/2 (skew symmetric)

Both Parts Can Be Used

From B to A is Not the Same as A to B (Gary Larson)

Asymmetries Are a Fact of Geography

Table of Mail Delivery Times Observe the asymmetry

A Map of Wind Computed from Mail Delivery Times

From Wind to Pressure Field

In the United States the Currency Indicates Where It Was Issued

Dollar Bill (Federal Reserve Note)

The 12 Federal Reserve Districts (Alaska and Hawaii omitted)

Movement of One Dollar Notes between Federal Reserve Districts, in hundreds, Feb. 1976 To: B NY P Cl R A Ch SL M K D SF

The Table of Dollar Bill Movements

Dollar Bill Movement in the U.S.

The Map is Computed Using a Continuous Version of the Gravity Model

First the Federal Reserve Districts Are “Rasterized”

Solving the Equations Yields the Potential Shown here by contours

Population movement in the United States

Nine Region Migration Table US Census 1973 (Note asymmetry)

The Population Change Information Can Be Positioned Locationally using centroids

Population Change at State Centroids

Gaining and Losing States Based on the marginals of a 48 by 48 migration table

The Conventional Net Movement Map Based on movement between state centroids (Computer sketch. Optimum deletion: values below mean ignored)

Using a model, this information can be converted to a potential field and its gradient

The Migration Potentials and Gradients

Migration Potentials and Streaklines

Recall that tens of millions of people are moving in the five year interval

That these migration maps resemble maps of wind or ocean currents is not surprising given that we in fact speak of migration flows and backwaters, and use many such hydrodynamic terms when discussing movement phenomena.

By the insertion of arbitrary areal boundaries, and by measuring the amount of flux across these boundaries, one can obtain information not contained in the original data, i.e., make a prediction.

Major Flux Across State Boundaries Predicted from the model and table marginals

The Previous Maps Have Used Observations based on States Patterns within urban areas could not be seen.

If we used the 3,141 counties of the United States the migration table could contain 9,862,740 numbers

County Units

The 9x106 numbers in a county to county table could not be comprehended without some vizualization techniques or without a model.

36,000 Communes of France

For a world table of international migration, refugee movements, commodity trade, one would have a table of nearly 40,000 entries.

Migration in Switzerland At several levels of resolution (Guido Dorigo, University of Zürich)

Net Migration In Switzerland By 3090 Gemeinde, 3.5 km resolution

Swiss Migration at Reduced Resolution To emphasize the filtering effect of resolution

In this presentation I have emphasized Location and Movement

I appreciate your attention and thank you.

Author: WALDO TOBLER